The polynomial given in the image is:

$P(x) = x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3$

To find the real zeros of this polynomial, we can apply methods such as factoring, synthetic division, or numerical approaches like the Rational Root Theorem, or using a calculator to approximate roots.

Let me analyze the polynomial and find the real roots. I'll calculate this for you.The real zeros of the polynomial $P(x) = x^5 - x^4 - 6x^3 + 14x^2 - 11x + 3$ are approximately:

$-3, 1, 1$

Note that the root $1$ has a multiplicity of two (it appears twice).

Would you like more details or explanations? Here are some related questions for further exploration:

1. What is the significance of repeated roots in a polynomial?
2. How can the Rational Root Theorem be applied to this problem?
3. How would synthetic division work for this polynomial?
4. What are the steps for factoring a polynomial with repeated roots?
5. How are complex roots identified when solving higher-degree polynomials?

Tip: If a root has multiplicity greater than 1, it means the graph of the polynomial touches the x-axis at that point but does not cross it.